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Edwin Flikkema, ICMS, Edinburgh, March 2012 From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK.

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Presentation on theme: "Edwin Flikkema, ICMS, Edinburgh, March 2012 From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK."— Presentation transcript:

1 Edwin Flikkema, ICMS, Edinburgh, March 2012 From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK

2 Edwin Flikkema, ICMS, Edinburgh, March 2012 Introduction and overview l Introduction: The minimal perimeter problem for 2D equal area bubble clusters. Systems of interacting particles Global optimisation l 2D particle clusters to 2D bubble clusters Voronoi construction l 2D particle systems: -log(r) or 1/r p repulsive potential Harmonic or polygonal confining potentials l Results

3 Edwin Flikkema, ICMS, Edinburgh, March D bubble clusters Minimal perimeter problem: l 2D cluster of N bubbles. l All bubbles have equal area. l Free or confined to the interior of a circle or polygon. l Minimize total perimeter (internal + external). Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.

4 Edwin Flikkema, ICMS, Edinburgh, March 2012 Systems of interacting particles System energy: Usually: Example: Lennard-Jones potential: LJ13: Ar 13 Used in: Molecular Dynamics, Monte Carlo, Energy landscapes

5 Edwin Flikkema, ICMS, Edinburgh, March 2012 Energy landscapes Stationary points of U: zero net force on each particle Minima of U correspond to (meta-)stable states. Global minimum is the most stable state. Local optimisation (finding a nearby minimum) relatively easy: Steepest descent, L-BFGS, Powell, etc. Global optimisation: hard. Energy vs coordinate Local optimisation

6 Edwin Flikkema, ICMS, Edinburgh, March 2012 Global optimisation methods Inspired by simulated annealing: l Basin hopping l Minima hopping Evolutionary algorithms: l Genetic algorithm Other: l Covariance matrix adaption l Simply starting from many random geometries

7 Edwin Flikkema, ICMS, Edinburgh, March D particle systems Energy: Repulsive inter-particle potential: Confining potential: or harmonic polygonal

8 Edwin Flikkema, ICMS, Edinburgh, March D particle clusters Pictures of particle clusters: e.g. N=41, bottom 3 in energy

9 Edwin Flikkema, ICMS, Edinburgh, March 2012 Particles to bubbles particle cluster Voronoi cells optimized perimeter Qhull Surface Evolver

10 Edwin Flikkema, ICMS, Edinburgh, March D particle clusters Polygonal confining potential: e.g. triangular contour lines discontinuous gradient: smoothing needed? unit vectors

11 Edwin Flikkema, ICMS, Edinburgh, March 2012 Technical details List of unique 2D geometries produced Problem: permutational isomers. Distinguishing by energy U not sufficient: l Spectrum of inter-particle distances compared. Gradient-based local optimisers have difficulty with polygonal potential due to discontinuous gradient l Smoothing needed? l Use gradient-less optimisers (e.g. Powell)?

12 Edwin Flikkema, ICMS, Edinburgh, March 2012 N=31-37 Results: bubble clusters: Free, circle, hexagon

13 Edwin Flikkema, ICMS, Edinburgh, March 2012 N=31-37 Results: bubble clusters: pentagon, square, triangle Elec. J. Combinatorics 17:R45 (2010)

14 Edwin Flikkema, ICMS, Edinburgh, March 2012 ConclusionsConclusions Optimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem. Various potentials have been tried. 1/r seems to work slightly better than –log(r). Using multiple potentials is recommended. Polygonal potentials have been introduced to represent confinement to a polygon

15 Edwin Flikkema, ICMS, Edinburgh, March 2012 AcknowledgementsAcknowledgements Simon Cox Adil Mughal

16 Edwin Flikkema, ICMS, Edinburgh, March 2012 Energy landscapes Stationary points of U: zero net force on each particle Minima of U correspond to (meta-)stable states. Global minimum is the most stable state. Saddle points (first order): transition states Network of minima connected by transition states Local optimisation (finding a nearby minimum) relatively easy: L-BFGS, Powell, etc. Global optimisation: hard. Energy vs coordinate Local optimisation

17 Edwin Flikkema, ICMS, Edinburgh, March D clusters: perimeter is fit to data for free clusters


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